As stated before, in the score competition the player has to keep his race in a fixed time \(T>0\)  visiting as many checkpoints as it is possible. This changes the model that has to be time dependent. Actually, as shown in [1]}, [2]}, the time dependent case can be seen as a special case of the stationary one, through a duplication of variables procedure. Therefore, the theoretical results valid for the stationary case, can be generally adapted to this new case. We limit ourself to report the form of the cost functional that is
\(J(t,x,q;\mathcal {S}) := \sum _{i=0}^{k}\int _{t_i}^{t^{i+1}}e^{-\lambda (s-t)} ds \\+ \sum _{i=1}^{k} e^{-\lambda (t_i-t)}C\left(y_{x,q}(t_i),Q(t_i^-),Q(t_i^+)\right),\) 
